Imaging using a multifocal aspheric lens to obtain extended depth of field

ABSTRACT

A system for imaging with a circularly symmetric multifocal aspheric lens is provided for obtaining an extended depth of field. The system includes a camera for capturing an image of at least one object through a circularly symmetric multifocal aspheric lens to provide a blurred image, and a computer system for processing the captured blurred image to provide a recovered image of the object having an extended depth of field. The recovered image may be outputted to a display or other peripheral device. Processing of the blurred image utilizes one of inverse filtering, convolution matrix (e.g., edge sharpening matrix), or maximum entropy. The computer system performing image processing may be in the camera or represent a computer system external to the camera which receives the blurred image. The extended depth of field is characterized by the object being in focus over a range of distances in the recovered image.

This application claims the benefit of priority to U.S. ProvisionalPatent Application No. 60/341,580, filed Dec. 18, 2001, which is hereinincorporated by reference.

The U.S. Government has rights in this invention pursuant to grant no.DAAD 19-00-1-0551 from U.S. Department of Defense/U.S. Army ResearchOffice.

FIELD OF THE INVENTION

The present invention relates to a system, method, and apparatus forimaging using a multifocal aspheric lens to obtain extended depth offield, and in particular to a system, method, and apparatus using acircularly symmetric multifocal aspheric lens to obtain a blurred imageand then processing of the blurred image to provide a recovered imagehaving an extended depth of field over which object or objects in theimage are in focus. The present invention also relates to a new class oflenses having a logarithmic phase function, which are circularlysymmetric, multifocal, and aspheric.

BACKGROUND OF THE INVENTION

In conventional digital camera photography, object or objects in animage of a scene are in focus at one distance (or distance range) fromthe camera often results in other objects at other distances in the samescene being out of focus. This is especially the case when imagedobjects are at different distances close to the camera, such as within10 feet or less, where optimal focus may be limited to a single limiteddistance range. Such conventional digital cameras may have a focusingmechanism to change the limited distance range where objects in theimage will be in focus. However, the focusing mechanism does not preventobjects outside this distance range being out of focus in the image.Thus, it would be desirable to provide imaging having an extended depthof field where the same object extending over a range of distances, ordifferent objects at different distances are all in focus in an image ofa common scene captured by a digital camera.

Prior research has developed optical systems for extending the depth offield either by the use of an apodization filter or by computerprocessing of purposefully blurred images, such as described in thefollowing academic literature: J. Ojeda-Castaneda, L. R. Berriel-Valdos,and E. Montes, Opt. Lett. 8, 458 (1983); T.-C. Poon, and M. Motamedi,Appl. Opt. 26, 4612 (1987); J. Ojeda-Castaneda, and L. R.Berriel-Valdos, Appl. Opt. 29, 994 (1990); E. R. Dowski, and W. T.Cathey, Appl. Opt. 34, 1859 (1995); J. van der Gracht, E. R. Dowski, W.T. Cathy and J. P. Bowen, Proc. SPIE 2537, 279 (1995); H. B. Wach, W. T.Cathey, and E. R. Dowski, Jr., Appl. Opt. 37, 5359 (1998); S. C. Tucker,E. R. Dowski, and W. T. Cathey, Optics Express 4, 467 (1999). Relatedresearch is also cited on axilenses which are optical elements thatconcentrate light energy along an optical axis, such as described in: L.M. Soroko, in Progress in Optics, E. Wolf, ed. (Elsevier, N.Y., 1989),pp109-160, and references therein; J. Sochacki, S. Bara, Z. Jaroszewicz,and A. Kolodziejczyk, Opt. Lett. 17, 7 (1992); J. Sochacki, A.Kolodziejczyk, Z. Jaroszewicz, and S. Bara, Appl. Opt. 31, 5326 (1992).

It is a feature of the present invention to capture images through acircularly symmetric multifocal aspheric lens providing a blurred imagewhich is then digitally processed to provide an image with an extendeddepth-of-field over which object or objects in the image are in focus.Prior approaches in extending depth of field described in theabove-identified literature have neither utilized a circularly symmetricaspheric lens, nor have provided processing of blurred images obtainedthrough such a lens to obtain images with improved focus over a largedepth of field.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an improved system,method, and apparatus for capturing and processing images to provide anextended depth of field using a circularly symmetric multifocal asphericlens.

It is another object of the present invention to provide an improvedsystem, method, and apparatus for capturing an image through acircularly symmetric multifocal lens to provide a blurred image andprocessing of the blurred image in accordance with the point spreadfunction of the lens to provide a recovered image in which one or moreobjects in a range of distances are in focus in accordance with themultifocal lengths of the lens.

It is a further object of the present invention that digital processingof such blurred images captured through a circular symmetric multifocallens may be carried out by various methods, such as inverse filtering,convolution matrix, or maximum entropy.

It is still another object of the present invention to provide a camerawith an extended depth of field to avoid the need for mechanicalfocusing required by prior art cameras to focus on objects at anyparticular distance.

Yet still another object of the present invention is to provide a newclass of optics having a logarithmic phase function for use in imagingapplications.

Briefly described, the system embodying the present invention includesan image capturing unit, such as a digital camera, having a circularlysymmetric aspheric lens (optics) to capture an image of one or morethree-dimensional objects in a scene, and an image processor, such as acomputer system, for processing the image to provide a recovered imagehaving an extended depth-of-field (or range of distances) over whichobject or objects in the image are in focus. The recovered image may beoutputted to a display or other peripheral device. The image processormay be part of the image capturing unit, or represent an externalcomputer system coupled to the display which receives the blurred image.Processing of the blurred image may be by one of inverse filter,convolution matrix (e.g., edge sharpening matrix), or maximum entropy inaccordance with the point spread function of the lens.

The circularly symmetric aspheric lens is multifocal in that its focallength varies continuously with the radius of the lens, in which thelens is characterized by the equation:${\phi(r)} = {- \left\{ {{\frac{2\pi}{\lambda_{0}}\left( {\sqrt{r^{2} + t^{2}} - t} \right)} + {\frac{\pi}{\lambda_{0}}\frac{R^{2}}{s_{2} - s_{1}}{\left. \quad\left\lbrack {{\ln\left\{ {{2{\frac{\quad{s_{2} - s_{1}}}{R^{2}}\left\lbrack {\sqrt{r^{2} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)^{2}} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)} \right\rbrack}} + 1} \right\}} - {\ln\left( {{4\frac{s_{2} - s_{1}}{R^{2}}s_{1}} + 1} \right)}} \right\rbrack \right\}.}}} \right.}$where, φ(r) is the phase delay for radius r of the lens to within anarbitrary constant, said range is over distances s₁ through s₂, R is theouter radius of the lens, t is the distance from the plane where thelens is disposed to the plane of image capture by said capturing means,and λ₀ is the free space wavelength.

The present invention utilizes a new class of lenses, called logarithmicaspheres. Different lenses of this class may be provided with differentextended depth of field performance in the above-described system byvarying the rate of change of focal length with radius, where eachdifferent lens has different phase delay logarithmic function φ(r), butare all circular symmetric and multifocal.

In an image-capturing unit representing a digital camera, theabove-described multifocal lens may replace the conventional(photographic) objective lens or lens system of the camera. Theabove-described multifocal lens may represent one or more opticalelements for multi-focal blurred imaging. For example, the multifocallens may represent a multi-focal phase plate (or mask), which may beused in combination with a conventional lens or lens system of a camera.This is particularly useful since such multi-focal phase plate can bereadily mounted on an existing camera to provide the above-describedimage-capturing unit and have an angular field of view in accordancewith the conventional lens of the camera. Although the lenses arediffraction limited, the system having a digital camera would not bediffraction limited due to its reliance on a CCD or other electronicimage detector.

In addition to the image capturing unit representing a digital camera(still or video), it may further represent a film-based camera forrecording on film the blurred image captured through the above-describedmultifocal lens, or conventional camera and phase plate, and then adigital scanner to digitize one of a print or negative representing theblurred image recorded on the film to provided a digitized blurredimage, in which the image processor receives and processes the digitizedblurred image to provide a recovered image.

The term object or objects may refer to any physical object, person, orother surroundings, in a scene, which may be located at one or moredistances, or extend over a range of distances, from the image capturingunit.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing objects, features, and advantages of the invention will bemore apparent from the following description in conjunction with thedrawings, in which:

FIG. 1 is a block diagram of the system in accordance with the presentinvention;

FIGS. 1A and 1B show is an example of a 3-dimensional objectrepresenting a staircase used to illustrate the system of FIG. 1, whereFIG. 1A is a perspective view of the staircase, and FIG. 1B shows adetailed view of each riser step of the staircase of FIG. 1A;

FIG. 1C is a block diagram showing the logarithmic asphere lens in thecamera of FIG. 1 where the lens is provided by multiple opticalelements;

FIG. 1D is a block diagram showing another embodiment of the camera ofFIG. 1 in which a phase plate is used in a cascade relationship with anobjective lens of a camera;

FIG. 2 is an optical diagram illustrating the notation for lens theoryin the system of FIG. 1;

FIG. 3 is an example illustrating a recovered image of the right side(letters of increasing size on each step with the steps at differentdistances) from the 3-dimensional object of FIG. 1A by the system ofFIG. 1;

FIG. 4A is a detailed view of part of the first step of the recoveredimage of FIG. 3;

FIG. 4B is an the image of the same part of the first step of the objectof FIG. 1A captured by the camera of FIG. 1 through a conventionalobjective lens rather than a multifocal lens in accordance with thepresent invention;

FIG. 5 is a graph of resolution (mm⁻¹) versus object distance (mm)illustrating the improved resolution of an imaged object by the systemof FIG. 1, as indicated by the solid line, and by a camera through aconventional objective lens of the same object, as indicated by thedashed line;

FIG. 6 illustrates the concept of maximum entropy processing which canbe used to recover the blurred image of the object captured by thecamera of FIG. 1;

FIGS. 7A, 7B, 7C, and 7D illustrate a computer simulation for thecomparison of the inverse filter and maximum entropy recovery processingfor the example of a two-point source object, where FIG. 7A shows animage of the two-point source object, FIG. 7B shows the blurred image ofthe two-point source object of FIG. 7A, FIG. 7C shows the resultingrecovered image by processing the blurred image of FIG. 7B by inversefiltering, and FIG. 7D shows the resulting recovered image by processingthe blurred image of FIG. 7B by maximum entropy;

FIG. 8 is a graph illustrating the pixel value for different pixelpositions along the same diagonal line through each of the images ofFIGS. 7B, 7C, and 7D, where pixels values of the line in the recoveredimage by maximum entropy (FIG. 7D) are indicated by a solid line, pixelsvalues of the line in the recovered image by inverse filtering (FIG. 7C)are indicated by a dashed line, and pixels values of the line in theblurred image (FIG. 7B) are indicated by a dotted line;

FIGS. 9A, 9B, 9C, and 9D illustrate a comparison of the inverse filterand maximum entropy recovery processing for the right part (letters) ofthe staircase object of FIG. 1A in the system of FIG. 1, where FIG. 9Ashows the blurred image of the object, and FIGS. 9B and 9C show theresulting recovered images by processing the blurred image of FIG. 9A byinverse filtering using different noise models for the Wiener-Helstromfilter labeled (A) and (B), and FIG. 9D shows the resulting recoveredimages by processing the blurred image of FIG. 9A by maximum entropy;

FIG. 10 is a graph illustrating a comparison of the sharpness of thepixels along a line through the recovered image provided by maximumentropy of FIG. 9D, as indicated by a dashed line, and the recoveredimage provided by inverse filter (A) of FIG. 9B, as indicated by a solidline, where the position of the line in each image is indicated, forexample, by the line in the blurred image of FIG. 9A;

FIG. 11 is a graph illustrating a comparison of the sharpness of thepixels along a line through the recovered image provided by maximumentropy of FIG. 9D, as indicated by a dashed line, and the recoveredimage provided by inverse filter (B) of FIG. 9C, as indicated by a solidline, where the position of the line in each image is indicated, forexample, by the line in the blurred image of FIG. 9A;

FIG. 11A is a more detailed view of graphs FIGS. 10 and 11 whencombined, in which sharpness of the pixels along the line through therecovered image of FIG. 9D (maximum entropy) is indicated by a dottedline, the recovered image of FIG. 9B (inverse filter (A)) is indicatedby a solid line, and the recovered image of FIG. 9C (inverse filter (B))is indicated by a dashed line;

FIGS. 12A, 12B, 12C and 12D show computer simulations of imagescomparing the different processes of image recovery upon a noisy blurredimage of FIG. 12A using edge sharpening filter in FIG. 12B, inversefilter in FIG. 12C, and maximum entropy in FIG. 12D; and

FIGS. 13A, 13B, 13C, and 13D are magnified images corresponding to asquare region of FIGS. 12A, 12B, 12C and 12D, respectively, in which theposition of the square region in each image is indicated by a box inFIG. 12A.

DETAILED DESCRIPTION OF THE INVENTION

Referring to FIGS. 1 and 1A, an integrated imaging and processing system8 is shown having a digital camera (or image capturing unit) 10 with aCCD array detector 11 for capturing images of an object 13 through amultifocal lens 14. The multifocal lens 14 represents a circularlysymmetric lens with a focal length that various continuously withradius, so that for a 3-dimensional object or objects, over a desireddistance range there is always an annular portion of the lens thatprovides a sharp in focus image, and the remainder of the lenscontributes to blurring. Such a lens 14 represents an aspheric lens, andis hereinafter referred to as logarithmic asphere lens 14, and wasdesigned using the Fermat's principle to find the transmission functionfor a lens that will image an extended portion of theobject-optical-axis into a single image point. For purposes ofillustration the object 13 is shown as a single three-dimensionalobject, such as a staircase, but object 13 may represent multipleobjects which may be located in a scene. The captured blurred image ofthe object 13 can represent a two dimensional array of pixels in whicheach of the pixels has a value depending on the resolution of the CCD ofthe camera. For example, 8, 16, or 32 bit resolution may be used. Eachcaptured blurred image of object 13 by camera 10 is outputted to animage processor representing a programmed microprocessor or computersystem 16 which processes the image and then outputs the processed imageto a display 18. Image processing may be provided by an inverse filter,or convolution matrix, applied to the pixels of the captured blurredimage from camera 10 to produce a recovered image of the object havingextended depth of field where the object is in focus in the image. Theconvolution matrix may be an edge sharpening filter. Other methods ofimage recovery may also be used, such as maximum entropy. The lens 14may represent one or more optical elements providing point-to-pointimaging and blurring, as shown in FIGS. 1 and 1C, or lens 14 mayrepresent a phase plate (or mask) 14 a (called herein a log-aspherephase plate) to provide blurring in a cascade relationship with anobjective lens 15 for imaging, as shown in FIG. 1D. For example, theobjective lens 15 for point to point imaging may be a conventionalphotographic lens. The lens 14 and phase plate 14 a is diffractionlimited in resolution. The design of lens 14 and digital processing ofimages captured through the lens is described below in more detail.

Computer system 16 may represent a personal computer, work station,lap-top, or other type of computer system, and the display 18 mayrepresent a CRT or LCD display. The computer system 16 also may storethe blurred and recovered images in memory, such as on a hard or opticaldisk, or output to other peripheral devices, such as a printer or vianetwork interface, such as modem, Ethernet, Internet, T1 line, or thelike, to other computer-based systems. Output of the captured image tocomputer system 16 may be through typical interface port (cable oroptical) used by conventional digital camera for transferring images (orimage files) to a computer system, or by storage of the captured imagein removable memory of the camera, such as memory card, memory chip,disk, PCMCIA card, and the like, such that the removable memory may beprovided to the separate computer system 16 for processing, via aninterface suitable reading the image from the removable memory.

Alternatively, the programmed microprocessor or computer system 16 (withor without display 18) may be part of the camera 10. Thus, system 8 canbe embodied on-board the housing of a digital camera having imagingthrough lens 14, where such camera provides the digital image processingof computer system 16.

Camera 10 may represent a typical digital camera adapted for use byreplacement of its objective lens with lens 14, as shown in FIG. 1 wherelens 14 represents a single element having a circularly symmetricaspheric body, or as shown in FIG. 1C where lens 14 represents multipleoptical elements, or as shown in FIG. 1D where lens 14 representslog-asphere phase plate 14 a in combination with the objective lens 15of the camera. Although two optical elements are shown in FIG. 1C, twoor more such elements can be used. The camera although described forstill images may be a digital video camera taking successive images inwhich each image is processed in near real-time by computer system 16.Although imaging is described for gray scale, color imaging may also beprovided by a suitable CCD array(s). One advantage of the digital camerawith lens 14 is that physical objects at different distances in animaged scene will, with image processing, be in focus in the outputtedimage. This is in contrast with a conventional digital camera withoutsystem 8 where an object in focus at one distance on the CCD array canresult in other objects at other distances in the same scene being outof focus. Although lens 14 or phase plate 14 a is diffraction limited, adigital camera utilizing such lens or plate would not be diffractionlimited in resolution due to the pixel size of the CCD array(s) or otherelectronic image detector(s).

The theory underlying the design of lens 14 and processing of capturedblurred images through this lens follows. In this discussion, an imageof an object (O) 13 is recorded (captured), such as the 3-dimensionalstaircase shown in FIG. 1, using camera 10 through aspherical lens (L)14, and the recorded blurred image is digitally processed by computersystem (DP) 16 and displayed on display (D) 18. An optical diagram ofFIG. 2 is first considered. Object points ranging over distances s₁through s₂ are all brought to focus at point P in plane (II) by means ofa continuous radial variation in the focal length. Dividing the lensinto annular rings of different focal lengths, one can verify that equalarea is obtained for each of N rings by choosing a radius r_(n) for then th ring as follows:r _(n)=(n/N)^(1/2) R,  (1)where R is the outer radius of the lens in plane (I).

Consider the imaging of point S at x(r_(n)) by the rays through theannular ring r_(n). To provide uniform or natural illumination, theinterval from s₁ to s₂ is subdivided into N segments; and the x(r_(n))segment is chosen to be weighted as follows:x(r _(n))=s ₁+(s ₂ −s ₁)n/N.  (2)Combining Eqs. (1) and (2) to eliminate the ratio n/N gives the basicequation for the lens, viz.,x(r)=s ₁+(s ₂ −s ₁)r ² /R ².  (3)As is well-known in physical optics, the general transmission function,t(r), for a lens can be written in the form:t(r)=exp[−iφ(r)],  (4)in which φ(r) is the phase delay. The form of the phase delay φ(r) canbe obtained by an application of Fermat's principle, see R. K. Luneburg,Mathematical Theory of Optics (university of California, 1964), p. 86.First, an expression for the total optical length L for the ray through(SOP) is written as follows:L={square root}{square root over (r ² +x ² )}+φ( r)λ₀/(2π)+{squareroot}{square root over (r ² +t ² )},  (5)where t is the distance from the lens plane (I) to (II) and λ₀ is thefree space wavelength.

From Fermat's principle and Eq. (5), setting ∂L/∂r=0 with x constant,and by Eq. (3), the following expression for the phase delay φ(r), viz.is found, $\begin{matrix}{{\phi(r)} = {{- \frac{2\pi}{\lambda_{0}}}{\int_{0}^{r}{\left\{ {\frac{r}{\sqrt{r^{2} + t^{2}}} + \frac{r}{\sqrt{r^{2} + \left\lbrack {s_{1} + {\left( {s_{2} - s_{1}} \right){r^{2}/R^{2}}}} \right\rbrack^{2}}}} \right\}\quad{{\mathbb{d}r}.}}}}} & (6)\end{matrix}$This can be directly integrated (see, for instance, H. B. Dwight, Tablesof Integrals and other Mathematical Data (Macmillan, New York, 1947) Eq.380.001, p. 70.) to yield the basic formula for the logarithmic aspherelens, expressed in two terms: $\begin{matrix}{{\phi(r)} = {- \left\{ {{\frac{2\pi}{\lambda_{0}}\left( {\sqrt{r^{2} + t^{2}} - t} \right)} + {\frac{\pi}{\lambda_{0}}\frac{R^{2}}{s_{2} - s_{1}}{\left. \quad\left\lbrack {{\ln\left\{ {{2{\frac{\quad{s_{2} - s_{1}}}{R^{2}}\left\lbrack {\sqrt{r^{2} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)^{2}} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)} \right\rbrack}} + 1} \right\}} - {\ln\left( {{4\frac{s_{2} - s_{1}}{R^{2}}s_{1}} + 1} \right)}} \right\rbrack \right\}.}}} \right.}} & (7)\end{matrix}$The first term is an ideal lens for point-to-point imaging with theobject point at infinity, and the second term introduces controlledblurring or aberration. As a first stage in the lens design, it isuseful to form a power series expansion of Eq. (7) using s₁=610 mm,s₂=6100 mm, R=5 mm, and t=25 mm, the first and second members of Eq. (7)are: $\begin{matrix}{{\phi(r)} = {{- \frac{2\pi}{\lambda_{0}}}{\left\{ {{0.02r^{2}} - {8 \times 10^{- 6}r^{4}} + {6.4 \times 10^{- 9}r^{6}} - {6.4 \times 10^{- 12}r^{8}} + {7 \times 10^{- 15}{r^{10}++}7.435 \times 10^{- 4}r^{2}} - {7.563 \times 10^{- 5}r^{4}} + {5.1431 \times 10^{- 6}r^{6}} - {1.803 \times 10^{- 7}r^{8}} + {2.4614 \times 10^{- 9}r^{10}}} \right\}.}}} & (8)\end{matrix}$These expansions are accurate to ±10⁻⁵ mm for the bracketed term. Theexpansions are obtained using NonlinearFit of Mathematica, which isdescribed in S. Wolfram, Mathematical Software 4.0 (Wolfram Research,Champaign, 2000) Statistics NonlinearFit, Sec. 3.8.1 (2000). Othervalues for s₁, s₂, R, and t may be used to provide a different φ(r)depending on the imaging application for the asphere lens.

Logarithmic asphere lens 14 may be being fabricated for t varying from24 mm to 80 nmm, but other values for t may be used. Recent advances inoptical fabrication methods make this type of logarithmic asphere lenspractical commercially. The logarithmic asphere lens may be fabricatedusing OptiPro Model SX50 computer-controlled asphere grinding machine,and a Magneto-Rheological Finisher commercially available from QEDTechnologies, LLC, Rochester, N.Y. The lenses may be fabricated in anoptical grade of quartz to an overall accuracy on the order of one-tenthwavelength. By measuring the point spread function for the logarithmicasphere lens of the camera, one can determine whether the lens is inagreement with theory to assure that it will properly provide a properlyblurred image for recovery by processing by the computer system 16. Asstated earlier, the fabricated lens 14 represents a circularly symmetriclens and is multifocal as the focal length of the lens variescontinuously with lens radius.

As described earlier, lens 14 may be provided by a two-stage opticalsystem in accordance with φ(r) of Equation 7, where the lens 14 isprovided by a conventional (photographic) lens 15 and amulti-focal-phase (corrector) plate 14 a, i.e., log-asphere phase plate,capable of providing a properly blurred image for recovery by processingof the computer system 16, as illustrated in the block diagram of FIG.1D. In other words, imaging and blurring is provided by two separatelenses, one lens 15 for ideal imaging and the other a phase plate 14 afor controlled blurring, rather than a single lens 14. For anylogarithmic asphere 14, there is a corresponding blurring phase-mask 14a. The phase plate 14 a may be fabricated using the same machinerymentioned above for making the asphere lens. The phase mask 14 a can beinserted into an optical system with any photographic lens 15 in orderto achieve an extended depth of field. For example, the double-Gausslens design commonly used in high-quality photographic lenses, the phaseplate may be located in the plane of the aperture stop. Essentially alllight rays passing this plane will thereby pass through the phase plateand contribute to the image. The phase plate 14 a may also be placed atother locations in the path of the light rays, but preferably is locatedat the aperture stop or at the exit plane of light into the camera.Hence, a cascade of a conventional imaging lens 15 with the phase plate14 a can provide an extended depth of field over the angular field ofview provided by the conventional imaging lens. For example, if lens 15has a wide-angle performance, this will be retained as a feature withextended depth of field.

An example showing the imaging and depth of field provided by thelogarithmic asphere lens is described below using a 3-dimensional objectwith 12 steps that are spaced axially by Δs of 50 mm, as shown inFIG. 1. The logarithmic asphere lens has a t=60 mm and the monochromeCCD array has a pixel size of 23 μm square which limits the basicresolution of the system. On the riser of each step, a strip resolutionchart is placed which contains a series of alphabet letters of varyingsize along with a chirped series of vertical lines, as shown in FIG. 1A.

Digital processing by the computer system 16 of the captured blurredimage of the object may be used to provide a recovered image in whichthe object is observable and in focus over a range of distance overwhich the object extends. One method for recovery of the blurred imageis to use an inverse filter or its equivalent matrix in picture space(in the image plane) based of the measured point spread function of thelens. Such an image plane matrix is used for convolution filtering. Thisfilter can be obtained by an inversion of the Fourier planeWiener-Helstrom inverse filter, such as described in B. R. Hunt, IEEETrans. Computer. C-22, 805 (1973), and R. C. Gonzalez, and R. E. Woods,Digital Image Processing (Addison-Wesley, 1992), p. 218. TheWiener-Helstrom inverse filter and its inverse filter in image space isshown below: WIENER-HELSTROM  INVERSE  FILTER${\hat{F}\left( {u,v} \right)} = {\frac{H^{*}\left( {u,v} \right)}{{{H\left( {u,v} \right)}}^{2} + {\gamma{{P\left( {u,v} \right)}}^{2}}}{G\left( {u,v} \right)}}$F̂(u, v)  recovered  spectrum G(u, v)  spectrum  of  blur  imageH(u, v)  transfer  function γP(u, v)²  noise  power  spectrum  densityγ  is  a  constant  determined  by  noise${{mean}\quad{and}\quad{{variance}.{P\left( {u,v} \right)}}} = {{FT}\left\{ {\begin{matrix}0 & {- 1} & 0 \\{- 1} & 4 & {- 1} \\0 & {- 1} & 0\end{matrix}} \right\}}$ INVERSE  FILTER  IN  IMAGE  SPACE:${{\hat{f}\left( {m,n} \right)} = {f^{- 1}\left\{ \frac{H^{*}\left( {u,v} \right)}{{{H\left( {u,v} \right)}}^{2} + {\gamma{{P\left( {u,v} \right)}}^{2}}} \right\}{{}_{}^{}{}_{}^{- 1}}\left\{ {G\left( {u,v} \right)} \right\}}}{FORM}\quad{THE}\quad{IMAGE}\quad{SPACE}\quad{CONVOLUTION}\quad{FILTER}$BY  APPROXIMATION

The convolution matrix applied to the blurred image may be a 5×5 matrixor a 3×3 matrix, such as shown, for example, below. The 3×3 matrix maybe very close to an edge-sharpening matrix. Hence, the filter need notbe strongly dependent on the point spread function. $\begin{matrix}{\begin{bmatrix}0 & 0 & {- 0.07} & 0 & 0 \\0 & 0.51 & {- 0.85} & 0.51 & 0 \\{- 0.07} & {- 0.85} & 5 & {- 0.85} & {- 0.07} \\0 & 0.51 & {- 0.85} & 0.51 & 0 \\0 & 0 & {- 0.07} & 0 & 0\end{bmatrix}\quad{{and}\quad\begin{bmatrix}0 & {- 1} & 0 \\{- 1} & 6 & {- 1} \\0 & {- 1} & 0\end{bmatrix}}} & (9)\end{matrix}$FIG. 3 shows a portion of the recovered (processed) 3-D step objectincluding 6 steps varying in object distance by 250 mm. The in-focusposition (Δ in FIG. 3) is at x=950 mm. FIG. 4A shows an enlargement ofthe final processed image at 30 in FIG. 3 that is 250 mm (5 steps)closer to the lens than the plane of best focus. For comparison in FIG.4B, the blurred image is shown using a Nikon 60 mm (objective) lensrather than the logarithmic asphere lens in the system 8 with the samef/D. This image is obtained using the same object positioning as thatpreviously described. A greatly extended depth of field is provided bythe logarithmic asphere lens.

In another example using the logarithmic asphere lens, the resolution ismeasured as a function of distance, as shown in FIG. 5. Data are takenat each step (50 mm) and the average value of 5 readings is plottedtogether as a solid line with error bars. The same setup as previouslydescribed is used except that resolution is measured using the fine-linechirped chart (left side of the staircase object of FIG. 1A). Also ineffect the pixel size has been reduced to 10 μm. The dotted line showsthe resolution limit for a 23 μm pixel size. For comparison, theresolution using the Nikon 60 mm (objective) lens, rather then theasphere lens, under identical conditions is also shown as a dashed linein FIG. 5. From these data, an increased depth-of-field for the camera10 with the logarithmic asphere lens 14 is shown.

Alternatively, the maximum entropy method may be programmed in computersystem 18 to recover the blurred image of the object 13 rather thanusing an inverse filter (or convolution filter). The maximum entropymethod is described, for example, in S. F. Gull and J. Skilling, MaximumEntropy Method In Image Processing, IEE Proc., Vol, 131, PT. F, No. 6,pp. 646-659 (1984). The basic process of the maximum entropy method isshown in FIG. 6. It represents an iterative process in which in eachcycle the difference is determined between the captured blurred image ofthe object and a calculated blurred image produced from an assumedobject convolved with the measured point source function. Thisdifference is used to change the assumed object for the next cycle, andso forth, until the difference is less than noise (or within a noisetolerance), at such time the assumed object represents the image of therecovered object.

The maximum entropy method can provide higher resolution images withless noise than the inverse filter method described earlier, asillustrated by the comparison images and graphs of FIGS. 7-13. In afirst example, FIG. 7A shows a two-point source object 13, and FIG. 7Bshow the computer simulated blurred image of the two-point object. Thecomputer system 16 processes the blurred image by inverse filtering toprovide the recovered image shown in FIG. 7C, and also processes theimage by maximum entropy to provide the recovered image shown in FIG.7D. FIG. 8 is a graph of pixel values by pixel position along a diagonalline through the two-point source object in each of the recovered imagesof FIGS. 7C and 7D to compare the two recovery methods of inversefiltering and maximum entropy. The diagonal line in each image isdenoted by the position of the white line 31 through the blurred image(FIG. 7B) of the two point source object 32 (as illustrated in the imageof FIG. 7B in the graph of FIG. 8). As this graph shows, maximum entropyyielded better results by providing an image with higher resolution andless noise than by inverse filtering, which magnified the noise. Forpurposes of comparison, the pixel values by pixel position along thediagonal line 31 in the blurred image of FIG. 7B is shown as a dottedline in FIG. 8.

In a second example, images captured and processed by the system 8 withrespect to the right part (letters) of the staircase object 13 of FIG.1A are shown in FIG. 9A-9D, where FIG. 9A shows the blurred noisy imagecaptured by camera 10. FIGS. 9B and 9C show recovered images of theblurred image processed by inverse filtering using two different noisemodels (A) and (B), respectively, while FIG. 9D shows the recoveredimage of the blurred image processed by maximum entropy. FIG. 10 aregraphs illustrating a comparison of the sharpness of the pixels along aline through the recovered image processed by maximum entropy of FIG. 9Dand by the inverse filter (A) of FIG. 9B. FIG. 11 are graphsillustrating a comparison of the sharpness of the pixels along a linethrough the recovered image processed by maximum entropy of FIG. 9D andby the inverse filter (B) of FIG. 9C. The position of this line in eachof the images analyzed in FIGS. 10 and 11 is indicated by white line 33in blurred image of FIG. 9A. Line 33 also appear below each of thegraphs of FIGS. 10 and 11. FIG. 1A is a more detailed view of part ofthe graphs of FIGS. 10 and 11 between pixel positions 162 through 178.In FIGS. 10, 11, and 11A, the bottom axis represent pixel position alongthe line, and the side axis represents sharpness as measured by theproportion of pixel value in the respective recovered image to the pixelvalue in the blurred image (FIG. 9A) at the same pixel position.

In a third example, computer simulations of a tiger image with a pointspread function of the logarithmic asphere lens are used to provide ablurred image of FIG. 12A, and recovered images are shown by processingusing an edge sharpening filter of FIG. 12B, an inverse filter of FIG.12C, and maximum entropy of FIG. 12D. FIGS. 13A-13D are magnified imagesof corresponding to a square region in each image FIGS. 12A-12D,respectively, of the same square region positioned in each image asillustrated by the white box outline 34 shown in FIG. 12A. This furtherillustrates that although different processing techniques may be appliedin system 8 to a captured blurred image to provide a recovered imagehaving an extended depth of field, the quality of the recovered imagemay depend on the processing method desired. Maximum entropy providessharper in focus images than an edge sharpening filter or inversefilter. However, one may select use of image plane filtering, edgesharpening or other convolution filter to improve image processingspeed.

One application of the circular-symmetric, multi-focal aspheric lens 14is to provide two particularly clear distances of operation, one is atarm's length, e.g., two feet, and the other at a longer distance, e.g.,20 feet to infinity. The camera 10 may be designed with a digital stillcamera or for a single use camera which will permit one to take theirown pictures with some scenery in the far background.

As described earlier, camera 10 with lens 14 may be used forconventional photography of scenery, family groups, and so on,consisting of the logarithmic asphere lens, CCD array 11, electronicsand computer processing means 16, which may be on-board the camera, orprocessed later. This system does not require mechanical focusing of aconventional camera, since within the extended depth of field (distancerange) characteristic of the particular lens 14, any object or subjectin the depth of field in photographs (images) will be in focus.

As stated earlier, lens 14 may be provided by a logarithmic phase plate14 a with any conventional (e.g., 35 mm) camera lens 15 (FIG. 1D). Thisphase plate may be mounted on any such conventional lens, just as oneuses UV filter or color bandpass filter in 35 mm photography, or atother locations in a camera, such as at the aperture stop. With thisphase plate mounted on a digital still camera, the output (blurredimage) of the CCD array is processed as described earlier in order toobtain extended depth of field. Thus, camera 10 may be a digital (CCD)still camera or video camera having one of asphere lens 14 or aconventional lens 15 and phase plate 14 a.

Although camera 10 is shown as using CCD array(s), other photodetectorarrays may be used, such as CMOS, CID, or the like. When camera 10represents a video camera, it is particularly useful since one can thenmake movies with greatly extended depth of field and at much lower lightlevels, since large aperture optics can be used. The resolution of thecamera 10, and generally of system 8, may be limited by the pixel-sizeof the CCD array of the camera, i.e., it is not diffraction limited.

Optionally, the system 8 may be used with camera 10 representing afilm-based camera having one of asphere lens 14 or a conventional lens15 and phase plate 14 a, as shown in FIG. 1C. The prints (or negatives)from such a film-based camera with the blurred image may then be scannedby a scanner into a digitized blurred image, which may be coupled to (ora file imported onto) the computer system 18, and then processed asdescribed earlier for a digitally captured blurred image to provide arecovered image. When film is used, the multifocal aspheric lens 14 isdiffraction limited, i.e., it provides an extended depth of field and itis diffraction limited as evidenced by the point spread function for themultifocal aspheric lens.

The system 8 provided by the present invention with the aspheric lens(or conventional lens and phase-plate) may be called a smart camera. Inphotography limited depth of field has been a great nuisance and it hasgreatly complicated camera design. In the smart camera, the picture(image) acquired at the CCD has been purposefully blurred and digitalimage processing can also be used for color correction. Examples havebeen described herein for a single logarithmic asphere lens, andseparately a phase mask (or plate) with a Nikon 60 mm lens. Imageprocessing results are shown comparing the Wiener-Helstrom inversefilter and maximum entropy methods; the latter providing better imagequality. Applications include digital video, DVD pickup unit, handheldlabel scanners, and single-use cameras, or other applications requiringextended depth of field imaging.

The logarithmic aspheric lens 14 described above represents a new classof lenses. Different lenses in the class are provided by changing theweighting of the ratio (r/R) in Equation 3 and subsequent Equations 6and 7 to effect rate of change of focal length to radius, thus providingdifferent phase delay functions φ(r) in the lens when fabricated. Eachsuch different lens can have different extended depth of fieldperformance in the above-described system 8, or in other imagingapplications.

From the foregoing description, it will be apparent that an improvedsystem, method, and apparatus for imaging is provided using alogarithmic multifocal aspheric lens, as well as a new class oflogarithmic multifocal aspheric lenses. Variations and modifications inthe herein described system, method, and apparatus will undoubtedlybecome apparent to those skilled in the art. Accordingly, the foregoingdescription should be taken as illustrative and not in a limiting sense.

1-65. (canceled)
 66. An imaging apparatus comprising: means forprocessing a blurred image, which was purposefully blurred by acircularly symmetric, multifocal lens, to provide a recovered image inaccordance with the point spread function of said lens.
 67. The imagingapparatus according to claim 66 wherein said processing mean utilizesone of an inverse filter, convolution matrix, or maximum entropy uponsaid blurred image to provide said recovered image.
 68. (canceled) 69.The optical element according to claim 70, wherein said body representsone of a lens and a phase-plate.
 70. An optical element comprising abody which is circularly symmetric, aspheric, and continuously varyingin focal length.
 71. A multifocal optical comprising a bodycharacterized by a phase function, wherein said phase function isprovided by φ(r) in accordance with the equation:${\phi(r)} = {- \left\{ {{\frac{2\pi}{\lambda_{0}}\left( {\sqrt{r^{2} + t^{2}} - t} \right)} + {\frac{\pi}{\lambda_{0}}\frac{R^{2}}{s_{2} - s_{1}}{\left. \quad\left\lbrack {{\ln\left\{ {{2{\frac{\quad{s_{2} - s_{1}}}{R^{2}}\left\lbrack {\sqrt{r^{2} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)^{2}} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)} \right\rbrack}} + 1} \right\}} - {\ln\left( {{4\frac{s_{2} - s_{1}}{R^{2}}s_{1}} + 1} \right)}} \right\rbrack \right\}.}}} \right.}$where, φ(r) is the phase delay for radius r of the lens to within anarbitrary constant, said range is over distances s₁ through s₂, R is theouter radius of the lens, t is the distance from the plane where thelens is disposed to the plane of image capture by said capturing means,and λ₀ is the free space wavelength.
 72. The optical element accordingto claim 70 wherein said body when imaged through provides enhancedviewability over a distance in accordance with said varying focallength.
 73. An imaging system comprising: a circularly symmetricmultifocal aspheric lens; and means for capturing an image through saidmultifocal aspheric lens.
 74. The system according to claim 73 whereinsaid lens provides an extended depth of field.
 75. The system accordingto claim 73 wherein said lens is characterized by the equation:${\phi(r)} = {- \left\{ {{\frac{2\pi}{\lambda_{0}}\left( {\sqrt{r^{2} + t^{2}} - t} \right)} + {\frac{\pi}{\lambda_{0}}\frac{R^{2}}{s_{2} - s_{1}}{\left. \quad\left\lbrack {{\ln\left\{ {{2{\frac{\quad{s_{2} - s_{1}}}{R^{2}}\left\lbrack {\sqrt{r^{2} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)^{2}} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)} \right\rbrack}} + 1} \right\}} - {\ln\left( {{4\frac{s_{2} - s_{1}}{R^{2}}s_{1}} + 1} \right)}} \right\rbrack \right\}.}}} \right.}$where, φ(r) is the phase delay for radius r of the lens to within anarbitrary constant, said range is over distances s₁ through s₂, R is theouter radius of the lens, t is the distance from the plane where thelens is disposed to the plane of image capture by said capturing means,and λ₀ is the free space wavelength.
 76. The system according to claim73 wherein said lens is provided by one optical element.
 77. The systemaccording to claim 73 wherein said lens is provided by an optical systemhaving multiple optical elements.
 78. The system according to claim 73wherein said capturing means comprises at least one photodetector arrayto capture said image through at least said lens.
 79. The systemaccording to claim 73 wherein said lens is characterized by alogarithmic phase function.
 80. A method for imaging comprising the stepof: imaging through a circularly symmetric, continuously varyingmultifocal aspheric lens to provide an image having an extended depth offield.
 81. The method according to claim 80 wherein said lens ischaracterized by the equation:${\phi(r)} = {- \left\{ {{\frac{2\pi}{\lambda_{0}}\left( {\sqrt{r^{2} + t^{2}} - t} \right)} + {\frac{\pi}{\lambda_{0}}\frac{R^{2}}{s_{2} - s_{1}}{\left. \quad\left\lbrack {{\ln\left\{ {{2{\frac{\quad{s_{2} - s_{1}}}{R^{2}}\left\lbrack {\sqrt{r^{2} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)^{2}} + \left( {s_{1} + {\frac{s_{2} - s_{1}}{R^{2}}r^{2}}} \right)} \right\rbrack}} + 1} \right\}} - {\ln\left( {{4\frac{s_{2} - s_{1}}{R^{2}}s_{1}} + 1} \right)}} \right\rbrack \right\}.}}} \right.}$where, φ(r) is the phase delay for radius r of the lens to within anarbitrary constant, said range is over distances s₁ through s₂, R is theouter radius of the lens, t is the distance from the plane where thelens is disposed to the plane of image capture by said capturing means,and λ₀ is the free space wavelength.
 82. An apparatus comprising: acircularly symmetric continuously varying multifocal aspheric lens; anda detector for receiving light through at least said lens.